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In this paper we are concerned with the integrability of the fourth Painlev� equation (PIV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlev� equation with parameters a�=�m,�b�=�−2(1�+�2n�+�m) where , is not integrable in the Liouville–Arnold sense by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and the Ziglin–Ramis–Morales-Ruiz–Sim� method yield the non-integrability results. [ABSTRACT FROM AUTHOR]